A q-DEFORMATION OF A TRIVIAL SYMMETRIC GROUP ACTION

Let A be the arrangement of hyperplanes consisting of the reflecting hyperplanes for the root system An−1. Let B = B(q) be the Varchenko matrix for this arrangement with all hyperplane parameters equal to q. We show that B is the matrix with rows and columns indexed by permutations with σ, τ entry equal to qi(στ −1) where i(π) is the number of inversions of π. Equivalently B is the matrix for left multiplication on CSn by Γn(q) = ∑ π∈Sn qπ. Clearly B commutes with the right-regular action of Sn on CSn. A general theorem of Varchenko applied in this special case shows that B is singular exactly when q is a j(j − 1)st root of 1 for some j between 2 and n. In this paper we prove two results which partially solve the problem (originally posed by Varchenko) of describing the Sn-module structure of the nullspace of B in the case that B is singular. Our first result is that ker(B) = indn Sn−1(Lien−1)/Lien in the case that q = e2πi/n(n−1) where Lien denotes the multilinear part of the free Lie algebra with n generators. Our second result gives an elegant formula for the determinant of B restricted to the virtual Sn-module P η with characteristic the power sum symmetric function pη(x).